Theorem i5lei4 350

Description: Relevance implication is less than or equal to non-tollens implication. (Contributed by NM, 26-Jun-2003.)

Assertion

Ref	Expression
i5lei4	(a →5 b) ≤ (a →4 b)

Proof of Theorem i5lei4

Step	Hyp	Ref	Expression
1		leo 158	. . . 4 a⊥ ≤ (a⊥ ∪ b)
2	1	leran 153	. . 3 (a⊥ ∩ b⊥ ) ≤ ((a⊥ ∪ b) ∩ b⊥ )
3	2	lelor 166	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ≤ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
4		df-i5 48	. 2 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
5		df-i4 47	. 2 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
6	3, 4, 5	le3tr1 140	1 (a →5 b) ≤ (a →4 b)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15   →₅ wi5 16

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i4 47  df-i5 48  df-le1 130  df-le2 131

This theorem is referenced by: (None)